Apollonius Circles

There are two completely different definitions of the so-called Apollonius circles:

1. The set of all points whose distances from two fixed points are in a constant ratio $1:\mu$ (Ogilvy 1990).

2. The eight Circles (two of which are nondegenerate) which solve Apollonius' Problem for three Circles.

Given one side of a Triangle and the ratio of the lengths of the other two sides, the Locus of the third Vertex is the Apollonius circle (of the first type) whose Center is on the extension of the given side. For a given Triangle, there are three circles of Apollonius.

Denote the three Apollonius circles (of the first type) of a Triangle by $k_1$, $k_2$, and $k_3$, and their centers $L_1$, $L_2$, and $L_3$. The center $L_1$ is the intersection of the side $A_2A_3$ with the tangent to the Circumcircle at $A_1$. $L_1$ is also the pole of the Symmedian Point $K$ with respect to Circumcircle. The centers $L_1$, $L_2$, and $L_3$ are Collinear on the Polar of $K$ with regard to its Circumcircle, called the Lemoine Line. The circle of Apollonius $k_1$ is also the locus of a point whose Pedal Triangle is Isosceles such that $\overline{P_1P_2}=\overline{P_1P_3}$.

Let $U$ and $V$ be points on the side line $BC$ of a Triangle $\Delta ABC$ met by the interior and exterior Angle Bisectors of Angles $A$. The Circle with Diameter $UV$ is called the $A$-Apollonian circle. Similarly, construct the $B$- and $C$-Apollonian circles. The Apollonian circles pass through the Vertices $A$, $B$, and $C$, and through the two Isodynamic Points $S$ and $S'$. The Vertices of the D-Triangle lie on the respective Apollonius circles.

See also Apollonius' Problem, Apollonius Pursuit Problem, Casey's Theorem, Hart's Theorem, Isodynamic Points, Soddy Circles

References

Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 40 and 294-299, 1929.

Ogilvy, C. S. Excursions in Geometry. New York: Dover, pp. 14-23, 1990.